# What does bar represent in $\frac{1}{\tau}\int_0^\tau \frac{dG}{dt}\equiv\frac{\overline{dG}}{dt}$?

What does bar represent on a differentiation? I had an equation like this

$$\frac{d}{dt}(\sum_i p_i\cdot r_i)=2T+\sum_i F_i \cdot r_i$$

Now they said to integrate it.

The time average of Eq. (3.24) over a time interval τ is obtained by integrating both sides with respect to t from 0 to τ , and dividing by τ :

Then I can see a bar on top of above equation.

$$\frac{1}{\tau}\int_0^\tau \frac{dG}{dt}\equiv\frac{\overline{dG}}{dt}=\overline{2T}+\overline{\sum_i F_i \cdot r_i}$$

Does the bar represent integration? I didn't ask why there's $$\frac{1}{\tau}$$ cause the book says they had divided by $$\tau$$ but my question is if we had divided by $$\tau$$ then why there's no $$\tau$$ in second and third term? (still I have the question)

• Bar means averaging here (averaging by time). Sep 22, 2021 at 9:40
• Time average of any quantity $A$ is $\bar{A}=\frac{1}{\tau}\int_0^\tau A dt$. Averages properties follow from integration properties. Sep 22, 2021 at 9:42

• The integration is from 0 to $\tau$ so they are dividing by $\frac{1}{\tau}- 0$ to calculate the averag (the "b- a" in what I wrote before. Sep 22, 2021 at 15:43